A332420 - OEIS

A332420

Number of Maclaurin polynomials p(2m-1,x) of sin(x) having exactly n positive zeros.

3, 4, 5, 4, 4, 4, 4, 5, 4, 4, 5, 4, 4, 4, 4, 5, 4, 4, 5, 4, 4, 4, 4, 5, 4, 4, 5, 4, 4, 4, 4, 5, 4, 4, 5, 4, 4, 5, 4, 4, 4, 4, 5, 4, 4, 5, 4, 4, 4, 4, 5, 4, 4, 5, 4, 4, 5, 4, 4, 4, 4, 5, 4, 4, 5, 4, 4, 5, 4, 4, 4, 4, 5, 4, 4, 5, 4, 4, 4, 4, 5, 4, 4, 5, 4, 4, 5, 4

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OFFSET

1,1

COMMENTS

Maclaurin polynomial p(2m-1,x) of sin(x) is x - x^3/3! + x^5/5! - ... - (-1)^m*x^(2m-1)/(2m-1)!.

LINKS

Table of n, a(n) for n=1..88.

EXAMPLE

a(1) counts these values of 2m-1: 3, 5, and 11. The single zeros of p(3,x), p(5,x), and p(11,x) are sqrt(6), 3.078642..., and 3.141148..., respectively.

MATHEMATICA

z = 60; p[n_, x_] := Normal[Series[Sin[x], {x, 0, n}]];

t[n_] := x /. NSolve[p[n, x] == 0, x, z];

u[n_] := Select[t[n], Im[#] == 0 && # > 0 &];

v = Table[Length[u[n]], {n, 2, 100, 2}]

Table[Count[v, n], {n, 1, 10}]

CROSSREFS

Cf. A012265, A332325.

Sequence in context: A275719 A014250 A155078 * A115051 A094634 A114545

Adjacent sequences: A332417 A332418 A332419 * A332421 A332422 A332423

KEYWORD

nonn

AUTHOR

Clark Kimberling, Feb 13 2020

EXTENSIONS

More terms from Jinyuan Wang, Jan 21 2025

STATUS

approved